Depth images dont get saved
If you just render pass. It should work as you described.
Possible reasons why it doesn't:

you don't have enabled Output Properties &gt; Postprocessing &gt; Pipeline &gt; Compositor
or you don't have mark checkbox on top row of Compositor Editor &gt; Use Nodes
or if you rendered only single image - these are not saved automatically (without add-on)


According to your screen, should be rendered only Depth pass.
To render and save final render (Composite node if enabled) and Depth pass (File Output node) at the same time on Render Animation action - node tree should look like this (for example):

Note: Composite node can be also replaced File Output node. Benefit is not only separate output file, but you can also specify different file format or directory.